A categoroid is defined by a “flexible" join of two categories, one over the usual objects and binary arrows, and the other over trigonoidal objects, or triples, and their associated morphisms. Recall that a category \({\cal C}\)  is defined by a collection of objects \({\cal O}\) , and a collection of morphisms \(f: a \rightarrow b\) , where \(a,b \in {\cal O}\) , each object \(c\)  is equipped with an identity morphism 1\(_c\) , and morphisms satisfy associative and compositional properties [1]}. Bridge arrows define the join between the two categories. Categoroids are used in this paper to analyze universal properties underlying conditional independence, illustrated by previous axiomatizations such as graphoids [2]}, [3]}, integer-valued multisets [4]}, [5]}, and separoids [6]}, [7]}. Categoroids are a type of category [1]}, with a novel dual parameterization, one involving the usual unary collection of objects and binary arrows mapping between them, but the other comprised of a collection of trigonoidal morphisms that act over triples of objects, with bridge morphisms defining the join. The trigonoidal morphisms represent a proof system defined over a conditional axiom system. Any valid conditional independence property can be inferred as a sequence of trigonoidal morphisms if such a property can be inferred by using the axiom system. Our primary interest is in developing a deeper understanding of how these two parameterizations interact. Figure REF  gives an overview of the different categoroids that will be discussed in this paper. The arrow indicates each categoroid in the co-domain is a special case of the categoroid in the domain. Not all relationships are shown to enhance readability.
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